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Vector Physics Notes Class 11 for CBSE, JEE and NEET: Addition, Components, Dot and Cross Product

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eSaral Academic and Editorial Team

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Summary

Vector physics notes Class 11 cover scalars and vectors, types of vectors, vector addition and subtraction, resolution into components, unit vectors, the dot product and the cross product. This page gives the key formulas, solved examples and handwritten notes for CBSE, JEE Main and NEET, and shows where vectors are used in mechanics and electromagnetism.

Vectors Physics Class 11 - IIT JEE | NEET

Vectors are not a separate chapter in the current NCERT Class 11 Physics book. They are introduced in Chapter 3, Motion in a Plane, and then used in almost every later chapter. The JEE Main 2026 syllabus also has no separate unit on vectors, but its units on Kinematics, Work, Energy and Power, and Electrostatics and Magnetism all need them. These vector physics notes Class 11 explain the operations once, so you can apply them in the later chapters.

Where Vector Physics Notes Class 11 Are Used

Vector concept

Where it appears again

Vector addition (triangle and parallelogram law)

Equilibrium of forces, Laws of Motion

Resolution into components

Projectile motion, inclined plane problems

Dot product

Work done by a force (W = F·d), power

Cross product

Torque (τ = r × F), angular momentum (L = r × p), magnetic force

Unit vectors (î, ĵ, k̂)

Three-dimensional problems in mechanics and electromagnetism

Relative velocity (vector subtraction)

Relative motion, river-boat problems

💡 Expert Tip by eSaral Academic Team, IIT Bombay Faculty:If you spend a few extra days on vectors now, later chapters become easier. Torque, work, and magnetic force questions often test vector operations in a disguised form.

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Scalars vs Vectors: Definition and Examples in Vector Physics Notes Class 11

Scalar Quantity

A scalar has only magnitude. Examples: mass, temperature, time, speed, distance, energy, pressure, work, power and electric current.

Vector Quantity

A vector has both magnitude and direction, and it follows the laws of vector addition. Examples: displacement, velocity, acceleration, force, momentum, torque, electric field and magnetic field.

Key Distinction

Distance is a scalar, and displacement is a vector. Speed is a scalar, and velocity is a vector. Some quantities have a direction but are still scalars, such as electric current and pressure, because they do not follow the vector addition rule.

A vector is written as A (bold) or with an arrow over the letter. Its magnitude is written as |A| or simply A, and it is always zero or positive.

Types of Vectors in Vector Physics Notes Class 11

Type

Definition

Example

Zero (null) vector

Magnitude 0, direction undefined

0

Unit vector

Magnitude 1, shows direction only

 = A/|A|

Equal vectors

Same magnitude and same direction

A = B

Negative vector

Same magnitude, opposite direction

−A

Parallel (collinear) vectors

Same or opposite direction, any magnitude

A ∥ B

Coplanar vectors

All lie in the same plane

Three vectors in the xy-plane

Position vector

Vector from the origin to a point P

r = xî + yĵ + zk̂

The standard unit vectors in three dimensions are î (along the x-axis), ĵ (along the y-axis) and k̂ (along the z-axis).

Addition and Subtraction of Vectors in Vector Physics Notes Class 11

Triangle Law of Vector Addition

If two vectors are drawn as two sides of a triangle in order, the third side, drawn in the reverse order, is their resultant: R = A + B.

Parallelogram Law of Vector Addition

If two vectors are the adjacent sides of a parallelogram, the diagonal that starts from the same point is their resultant.

  • Magnitude: R = √(A² + B² + 2AB cos θ), where θ is the angle between A and B.
  • Direction: the angle α of R with A is given by tan α = B sin θ / (A + B cos θ).

Special Cases

Angle θ between the vectors

Magnitude of the resultant

0° (same direction)

R = A + B (maximum)

180° (opposite direction)

R = |A − B| (minimum)

90°

R = √(A² + B²)

60°

R = √(A² + B² + AB)

120°

R = √(A² + B² − AB)

The magnitude of the resultant always lies between |A − B| and A + B.

Vector Subtraction

A − B = A + (−B). If θ is the angle between A and B, then |A − B| = √(A² + B² − 2AB cos θ).

Tip: In MCQs, use the special cases (0°, 90°, 180°) to remove wrong options quickly, without doing the full calculation.

Resolution of Vectors and Unit Vectors in Vector Physics Notes Class 11

Resolving a Vector into Components

A vector A in a plane can be resolved into two perpendicular components. If θ is the angle that A makes with the x-axis:

  • Aₓ = A cos θ (along x)
  • Aᵧ = A sin θ (along y)
  • A = √(Aₓ² + Aᵧ²) and θ = tan⁻¹(Aᵧ/Aₓ)

Three Dimensions

A = Aₓî + Aᵧĵ + A_z k̂, and |A| = √(Aₓ² + Aᵧ² + A_z²).

Unit Vector Formula

 = A/|A| = (Aₓî + Aᵧĵ + A_z k̂) / √(Aₓ² + Aᵧ² + A_z²). A unit vector has magnitude 1 and points in the direction of A.

Adding Vectors Using Components

  1. Resolve each vector into components.
  2. Add the components along each axis: R = (Aₓ + Bₓ)î + (Aᵧ + Bᵧ)ĵ.
  3. Find the magnitude and the direction of R.

This method works for any number of vectors, and it is the most reliable method for exam problems.

Dot Product (Scalar Product) in Vector Physics Notes Class 11

The dot product of two vectors is a scalar.

  • Definition: A·B = AB cos θ, where θ is the angle between the vectors.
  • Component form: A·B = AₓBₓ + AᵧBᵧ + A_zB_z.

Property

Result

A·A

A²

A·B = 0 (θ = 90°)

The vectors are perpendicular

î·î = ĵ·ĵ = k̂·k̂

1

î·ĵ = ĵ·k̂ = k̂·î

0

Commutative

A·B = B·A

Physics applications: work W = F·d = Fd cos θ, power P = F·v, and the angle between two vectors, cos θ = (A·B)/(AB).

Cross Product (Vector Product) in Vector Physics Notes Class 11

The cross product of two vectors is a vector that is perpendicular to both of them.

  • Definition: A × B = AB sin θ n̂, where n̂ is the unit vector perpendicular to both, found with the right-hand rule.
  • Component form: A × B = î(AᵧB_z − A_zBᵧ) − ĵ(AₓB_z − A_zBₓ) + k̂(AₓBᵧ − AᵧBₓ). This is the expansion of the determinant with the rows (î, ĵ, k̂), (Aₓ, Aᵧ, A_z) and (Bₓ, Bᵧ, B_z).

Property

Result

A × A

0

A × B = 0 (θ = 0° or 180°)

The vectors are parallel

î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ

Cyclic order

ĵ × î = −k̂

Reverse order gives a minus sign

|A × B|

AB sin θ, the area of the parallelogram

Anti-commutative

A × B = −(B × A)

Physics applications: torque τ = r × F, angular momentum L = r × p, and magnetic force F = q(v × B).

Key Formulas in Vector Physics Notes Class 11

Formula

Description

R = √(A² + B² + 2AB cos θ)

Resultant magnitude (parallelogram law)

tan α = B sin θ / (A + B cos θ)

Direction of the resultant from A

|A − B| = √(A² + B² − 2AB cos θ)

Magnitude of the difference

Aₓ = A cos θ, Aᵧ = A sin θ

Components

|A| = √(Aₓ² + Aᵧ² + A_z²)

Magnitude from components

 = A/|A|

Unit vector

A·B = AB cos θ = AₓBₓ + AᵧBᵧ + A_zB_z

Dot product

|A × B| = AB sin θ

Magnitude of the cross product

W = F·d

Work

τ = r × F

Torque

Solved Examples from Vector Physics Notes Class 11

Example 1 (resultant): Two forces of 3 N and 4 N act at a point. Find the resultant when the angle between them is (a) 90° and (b) 60°.

(a) R = √(3² + 4²) = 5 N. (b) R = √(9 + 16 + 2 × 3 × 4 × cos 60°) = √37 ≈ 6.08 N.

Example 2 (unit vector): Find the magnitude and the unit vector of A = 2î + 3ĵ + 6k̂.

|A| = √(4 + 9 + 36) = 7, so  = (2î + 3ĵ + 6k̂)/7.

Example 3 (dot product and angle): Find the angle between A = 2î + 3ĵ − k̂ and B = î − 2ĵ + 4k̂.

A·B = 2 − 6 − 4 = −8. |A| = √14 and |B| = √21, so cos θ = −8/√294 ≈ −0.467, and θ ≈ 118°.

Example 4 (cross product): Find A × B for A = î + 2ĵ + 3k̂ and B = 2î − ĵ + k̂.

A × B = î(2 × 1 − 3 × (−1)) − ĵ(1 × 1 − 3 × 2) + k̂(1 × (−1) − 2 × 2) = 5î + 5ĵ − 5k̂. Its magnitude is 5√3 ≈ 8.66.

Example 5 (work): A force F = (3î + 4ĵ) N moves an object through a displacement d = (5î + 2ĵ) m. Find the work done.

W = F·d = 3 × 5 + 4 × 2 = 23 J.

Common Mistakes in Vector Physics Notes Class 11 Problems

These are the errors that students make most often when they apply vector physics notes Class 11 to problems.

Mistake

Correct approach

Adding magnitudes instead of vectors

Use the parallelogram law or components. Add magnitudes only when the vectors point the same way.

Using sin instead of cos in the dot product

Dot product: cos θ. Cross product: sin θ.

Forgetting that the cross product is not commutative

A × B = −(B × A)

Taking the angle between vectors from the wrong end

Draw both vectors from the same point before you find θ

Treating current or pressure as a vector

They have a direction but follow scalar addition

Writing a negative magnitude

The magnitude of a vector is never negative

How to Study Vector Physics Notes Class 11 for Exams

  1. Learn the operations of the vector physics notes Class 11 in this order: addition and subtraction, components, dot product, cross product.
  2. Practise writing vectors in î, ĵ, k̂ form, because most calculations are easier in that form.
  3. Apply the ideas to motion with eSaral’s vectors in two dimensional motion page, which covers projectile and circular motion.
  4. For textbook questions, use eSaral’s NCERT Solutions for Class 11 Physics Chapter 3.
  5. For harder vector-based problems at the JEE Advanced level, solve the selected problems from our Irodov questions for JEE collection.

Frequently Asked Questions

What is a vector in Physics Class 11?

A vector is a quantity with both magnitude and direction that follows the laws of vector addition, such as displacement, velocity, force, momentum and electric field. In these vector physics notes Class 11, a vector is written in bold or with an arrow, and its magnitude is never negative. Scalars such as mass, temperature and speed have only magnitude.

What is the difference between dot product and cross product?

The dot product, A·B = AB cos θ, gives a scalar. It is used for work (W = F·d) and power. The cross product, A × B = AB sin θ n̂, gives a vector perpendicular to both. It is used for torque (τ = r × F) and magnetic force. The dot product is maximum when the vectors are parallel, and the cross product is maximum when they are perpendicular.

Is Vectors a chapter in the Class 11 Physics NCERT book?

No. In the current NCERT Class 11 Physics book, vectors are introduced in Chapter 3, Motion in a Plane. They are then used in the chapters on Laws of Motion, Work, Energy and Power, Rotational Motion and others. These vector physics notes Class 11 on eSaral cover the topic in one place.

How many marks does Vectors carry in JEE Main?

Vectors is not listed as a separate unit in the JEE Main 2026 syllabus, so there is no fixed number of marks. Vector operations are used in Kinematics, Work, Energy and Power, Rotational Motion, Electrostatics and Magnetism questions. Practise previous year questions to see how vectors are used in each unit.

What is the Triangle Law of Vector Addition?

In vector physics notes Class 11, if two vectors are drawn as two sides of a triangle in order, the third side drawn in the reverse order represents their resultant. For the magnitude, R = √(A² + B² + 2AB cos θ), where θ is the angle between the two vectors when they are drawn from the same point. It is equivalent to the parallelogram law.

What are the standard unit vectors?

In vector physics notes Class 11, the three standard unit vectors are î (x-axis), ĵ (y-axis) and k̂ (z-axis), and each has magnitude 1. Any vector in three dimensions can be written as A = Aₓî + Aᵧĵ + A_zk̂. The cross products are î × ĵ = k̂, ĵ × k̂ = î and k̂ × î = ĵ, and the dot product of any two different unit vectors is zero.

How do you find the angle between two vectors?

Use the dot product: cos θ = (A·B)/(AB), where A·B = AₓBₓ + AᵧBᵧ + A_zB_z. For example, vectors with a zero dot product are perpendicular. This method is part of the vector physics notes Class 11 formulas on this page.

How do you resolve a vector into components?

In vector physics notes Class 11, if a vector A makes an angle θ with the x-axis, its components are Aₓ = A cos θ and Aᵧ = A sin θ, and A = √(Aₓ² + Aᵧ²). You can add any number of vectors by adding their components along each axis.

Where can I find more Class 11 Physics notes on eSaral?

eSaral has chapter-wise Class 11 Physics notes, NCERT solutions, handwritten notes on topics such as work, energy and power and circular motion, and the vector physics notes Class 11 on this page, so you can study the topic and revise it in one place.

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eSaral Academic and Editorial Team

Team eSaral is the collective author profile for educational content created by eSaral’s teachers and academic contributors. The team draws on expertise from IIT graduates, doctors, experienced educators and subject specialists to develop resources for JEE, NEET and school students. Our articles aim to explain concepts clearly and help students study with confidence.

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